SAMPLING THEOREM COMBINED WITH AN EFFICIENT RECONSTRUCTION ALGORITHM, HAS BEEN APPLIED SUCCESSFULLY TO PROVE IRREGULAR SAMPLING THEOREMS IN SPACES OF ANALYTIC FUNCTIONS, FOR SHORTTIME FOURIER TRANSFORMS AND WAVELET TRANSFORMS, AND FOR A GENERAL CLASSOF SPACES OF BAND-LIMITED FUNCTIONS.AN IMPORTANT INSTANCE OF THIS ALGORITHM OCCURS IN THE PRESENCE OF FrameS IN A HILBERT SPACE.WE PRESENT A NEWMETHOD TO OBTAIN CONTINUOUS Frame BOUNDS. IN THE APPLICATIONS TO IRREGULAR SAMPLING OF BAND-LIMITED FUNCTIONS THIS ALTERNATIVE STRATEGY LEADS TO BETTER AND EXPLICIT ESTIMATES OF THE CONTINUOUS Frame BOUNDS.

CERTAIN FACTS ABOUT FrameS ARE EXTENDED FOR THE DIFFERENT KINDS OF FrameS IN HILBERT C*-MODULES. THE PAPER PRESENTS THE FUSION FrameS WITH C*-VALUED BOUNDS IN HILBERT C*-MODULES WHERE THEY ARE CALLED *-FUSION FrameS. ALSO, THE RELATIONS BETWEEN FUSION FrameS AND *-FUSION FrameS IN HILBERT C*-MODULES ARE GIVEN SUCH THAT *-FUSION FrameS CAN BE STUDIED AS FUSION FrameS WITH DIFFERENT BOUNDS.

The aim of this article is to discuss operator monotone and operator convex functions, introduce some operator inequalities which are proved by operator monotone and operator convex functions.

IN THIS PAPER, WE INTRODUCE THE NOTIONS OF operator (A, B, G) -MEAN, RELATIVE operator (A, B, G) -ENTROPY AND TSALLIS RELATIVE operator (A, B, G) -ENTROPY. WE GIVE UPPER AND LOWER BOUNDS OF RELATIVE operator (0, B, G) -ENTROPY AND TSALLIS RELATIVE operator (A, B, G) -ENTROPY WITH RESPECT TO operator (A, B, G) -MEAN.

WE GENERALIZED THE CONCEPT OF FUSION FrameS, NAMELY, C-FUSION INTEGRAL, WHICH IS A CONTINUOUS VERSION OF THE FUSION FrameS AND WE EXTENDED IT FOR GENERALIZED FrameS. IN THIS ARTICLE WEGIVE SOME IMPORTANT PROPERTIES ABOUT IT NAMELY ERASURES OF SUBSPACES, THE BOUND OF GC−ERASURERECONSTRUCTION ERROR FOR PARSEVAL GC-Frame OF SUBSPACES.